Voronoi polytopes for polyhedral norms on lattices

Abstract

A polyhedral norm is a norm N on Rn for which the set N(x)≤1 is a polytope. This covers the case of the L1 and L∞ norms. We consider here effective algorithms for determining the Voronoi polytope for such norms with a point set being a lattice. The algorithms, that we propose, use the symmetries effectively in order to compute a decomposition of the space into convex polytopes named VN-spaces. The Voronoi polytopes and other geometrical information are easily obtained from it.